let n1, n2, n3, n4, n5, n6 be Element of NAT ; for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
let S be Gene-Set; for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
let p1, p2 be Individual of S; ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
A1:
( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) )
proof
assume
n6 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5)
then
n6 >= len S
by Def1;
then A2:
n6 >= len (crossover (p1,p2,n1,n2,n3,n4,n5))
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6)
= crossover (
(crossover (p1,p2,n1,n2,n3,n4,n5)),
(crossover (p2,p1,n1,n2,n3,n4,n5)),
n6)
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6)
= crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
by A2, Th5;
verum
end;
A3:
( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) )
proof
assume A4:
n2 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6)
then
n2 >= len S
by Def1;
then A5:
n2 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6) =
crossover (
(crossover (p1,p2,n1,n3,n4,n5)),
(crossover (p2,p1,n1,n2,n3,n4,n5)),
n6)
by A4, Th46
.=
crossover (
(crossover (p1,p2,n1,n3,n4,n5)),
(crossover (p2,p1,n1,n3,n4,n5)),
n6)
by A5, Th46
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6)
= crossover (
p1,
p2,
n1,
n3,
n4,
n5,
n6)
;
verum
end;
A6:
( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) )
proof
assume A7:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6)
then
n5 >= len S
by Def1;
then A8:
n5 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6) =
crossover (
(crossover (p1,p2,n1,n2,n3,n4)),
(crossover (p2,p1,n1,n2,n3,n4,n5)),
n6)
by A7, Th46
.=
crossover (
(crossover (p1,p2,n1,n2,n3,n4)),
(crossover (p2,p1,n1,n2,n3,n4)),
n6)
by A8, Th46
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6)
= crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n6)
;
verum
end;
A9:
( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) )
proof
assume A10:
n4 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6)
then
n4 >= len S
by Def1;
then A11:
n4 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6) =
crossover (
(crossover (p1,p2,n1,n2,n3,n5)),
(crossover (p2,p1,n1,n2,n3,n4,n5)),
n6)
by A10, Th46
.=
crossover (
(crossover (p1,p2,n1,n2,n3,n5)),
(crossover (p2,p1,n1,n2,n3,n5)),
n6)
by A11, Th46
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6)
= crossover (
p1,
p2,
n1,
n2,
n3,
n5,
n6)
;
verum
end;
A12:
( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) )
proof
assume A13:
n3 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6)
then
n3 >= len S
by Def1;
then A14:
n3 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6) =
crossover (
(crossover (p1,p2,n1,n2,n4,n5)),
(crossover (p2,p1,n1,n2,n3,n4,n5)),
n6)
by A13, Th46
.=
crossover (
(crossover (p1,p2,n1,n2,n4,n5)),
(crossover (p2,p1,n1,n2,n4,n5)),
n6)
by A14, Th46
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6)
= crossover (
p1,
p2,
n1,
n2,
n4,
n5,
n6)
;
verum
end;
( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) )
proof
assume A15:
n1 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6)
then
n1 >= len S
by Def1;
then A16:
n1 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6) =
crossover (
(crossover (p1,p2,n2,n3,n4,n5)),
(crossover (p2,p1,n1,n2,n3,n4,n5)),
n6)
by A15, Th46
.=
crossover (
(crossover (p1,p2,n2,n3,n4,n5)),
(crossover (p2,p1,n2,n3,n4,n5)),
n6)
by A16, Th46
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5,
n6)
= crossover (
p1,
p2,
n2,
n3,
n4,
n5,
n6)
;
verum
end;
hence
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
by A3, A12, A9, A6, A1; verum